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Theory and Methods New Properties of the Kumaraswamy Distribution This article was downloaded by: [Stanford University] On: 27 November 2014, At: 20:23 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK Communications in Statistics - Theory and Methods Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/lsta20 New Properties of the Kumaraswamy Distribution Pablo A. Mitnik a a Center on Poverty and Inequality , Stanford University , Stanford , California , USA Published online: 22 Jan 2013. To cite this article: Pablo A. Mitnik (2013) New Properties of the Kumaraswamy Distribution, Communications in Statistics - Theory and Methods, 42:5, 741-755, DOI: 10.1080/03610926.2011.581782 To link to this article: http://dx.doi.org/10.1080/03610926.2011.581782 PLEASE SCROLL DOWN FOR ARTICLE Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) contained in the publications on our platform. However, Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information. Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content. This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. Terms & Conditions of access and use can be found at http:// www.tandfonline.com/page/terms-and-conditions Communications in Statistics—Theory and Methods, 42: 741–755, 2013 Copyright © Taylor & Francis Group, LLC ISSN: 0361-0926 print/1532-415X online DOI: 10.1080/03610926.2011.581782 New Properties of the Kumaraswamy Distribution PABLO A. MITNIK Center on Poverty and Inequality, Stanford University, Stanford, California, USA The Kumaraswamy distribution is very similar to the Beta distribution but has the key advantage of a closed-form cumulative distribution function. This makes it much better suited than the Beta distribution for computation-intensive activities like simulation modeling and the estimation of models by simulation-based methods. However, in spite of the fact that the Kumaraswamy distribution was introduced in 1980, further theoretical research on the distribution was not developed until very recently (Garg, 2008; Jones, 2009; Mitnik, 2009; Nadarajah, 2008). This article contributes to this recent research and: (a) shows that Kumaraswamy variables exhibit closeness under exponentiation and under linear transformation; (b) derives an expression for the moments of the general form of the distribution; (c) specifies some of the distribution’s limiting distributions; and (d) introduces an analytical expression for the mean absolute deviation around the median as a function of the parameters of the distribution, and establishes some bounds for this dispersion measure and for the variance. Keywords Beta distribution; Closeness properties; Dispersion bounds; Kumaraswamy distribution; Limiting distributions; Mean absolute deviation around the median; Moments. Mathematics Subject Classification Primary 60E05; Secondary 62E99. 1. Introduction Downloaded by [Stanford University] at 20:23 27 November 2014 The Kumaraswamy distribution is a continuous probability distribution with double-bounded support. It is very similar to the Beta distribution, and can thus assume a strikingly large variety of shapes and be used to model many random processes and uncertainties. One key difference between the Kumaraswamy and Beta distributions is the availability for the former, but not for the latter, of an invertible closed-form cumulative distribution function. This makes it much better suited than the Beta distribution for computation-intensive activities like simulation modeling (e.g., Banks, 1998; Rennard, 2006) and the estimation of models by simulation-based methods (e.g., the method of simulated moments and Received May 16, 2008; Accepted April 13, 2011 Address correspondence to Pablo A. Mitnik, Center on Poverty and Inequality, Stanford University, 450 Serra Mall, Building 370, Room 212, Stanford, CA 94305-2077, USA; E-mail: [email protected] 741 742 Mitnik indirect inference; see, for instance, Gouriéroux and Monfort, 1996). In these activities, which have become increasingly important in the last 15 years, using the Kumaraswamy rather than the Beta distribution would be much more efficient and easier to implement from a computational point of view. This is of paramount importance in settings like these, where computer-power constraints are often binding. In addition, the Kumaraswamy distribution can be fruitfully employed in the context of the quantile modeling approach (Jones, 2009, p. 71) and, in particular, to model conditional quantiles parametrically (Mitnik, 2009, Sec. 4). The Kumaraswamy distribution was originally conceived to model hydrological phenomena (Kumaraswamy, 1980), and has been used for this but also for other purposes (for examples see Courard-Hauri, 2007; Fletcher and Ponnambalam, 1996; Ganji et al., 2006; Sanchez et al., 2007; Seifi et al., 2000; Sundar and Subbiah, 1989). However, in spite of the advantages that the availability of a closed- form distribution function entails, and although Poondi Kumaraswamy introduced the double-bounded distribution that bears his name almost three decades ago, further research on the distribution itself was not developed until very recently (Garg, 2008; Nadarajah, 2008; Mitnik, 2009; and, most notably, Jones, 2009). The results presented here in—a by-product of on-going research on queuing matching models of the labor market and on parametric quantile regression, in which the Kumaraswamy distribution is employed intensively—contribute to this recent research by uncovering several new properties of this distribution. The article is organized as follows. Section 2 presents the features of the Kumaraswamy distribution relevant for the rest of the article—all of which were essentially contained in Kumaraswamy’s 1980 article—and very briefly reviews the recent literature on this distribution. Section 3 introduces two very simple but previously unreported properties of Kumaraswamy variables: closeness under linear transformation and under exponentiation. Section 4 derives an expression for the moments of the general form of the distribution. Section 5 specifies some of the distribution’s limiting distributions. Lastly, Sec. 6 introduces an analytical expression for the mean absolute deviation around the median as a function of the parameters of the distribution, and establishes some bounds both for this dispersion measure and for the variance. 2. The Kumaraswamy Distribution: Definition and Known Properties In its general form, the probability density function of the continuous part of the Downloaded by [Stanford University] at 20:23 27 November 2014 distribution Kumaraswamy introduced in his 1980 article can be written as 1 z − c p−1 z − c p q−1 f z = pq 1 − c<z<b (1) Z b − c b − c b − c with shape parameters p>0 and q>0, and boundary parameters c and b. The general form of the distribution will be denoted by Kp q c b. Making the = Z−c transformation X b−c and using the change of variable theorem, we obtain the standard form of the Kumaraswamy density function = p−1 − p q−1 fXx pqx 1 x 0 <x<1 (2) which will be denoted by Kp q ≡ Kp q 0 1. In what follows the standard form of the distribution will be employed unless otherwise indicated. The Kumaraswamy Distribution 743 The cumulative distribution function of the Kumaraswamy distribution has a closed form expression, namely Fx = 1 − 1 − xpq 0 <x<1 (3) From (3), it immediately follows that the quantile function F −1u is also available in closed-form: 1 1 x = 1 − 1 − u q p 0 <u<1 (4) In particular, the median of the Kumaraswamy distribution can be written as 1 1 mdX = = 1 − 05 q p (5) If the random variable X is distributed Kp q, its moments around zero can be expressed as r X = qB 1 + q (6) r p = 1 −1 − −1 = = where B s 1 s ds + is the Beta function and v 0 v−1 −t 0 t e dt is the Gamma function. Thus, the expectation and variance of X are 1 EX = = X = qB 1 + q (7) 1 p 2 1 2 VarX = = X − 2 = qB 1 + q − qB 1 + q (8) 2 2 p p Table 1 shows the possible shapes of the Kumaraswamy distribution and its behavior at the boundaries of its support, as a function of the values of its parameters; the table also identifies the distribution’s special cases. Notably, as also pointed out by Jones (2009, p. 74), the relationships between the features of the Kumaraswamy distribution and the values of its parameters summarized in Table 1 also obtain, without any exception, in the case of the Beta distribution (for the Beta Downloaded by [Stanford University] at 20:23 27 November 2014 distribution, see Johnson et al., 1995, p. 219). As anticipated in the Introduction, recent research has substantially extended our knowledge of the Kumaraswamy distribution’s properties and of its relationships with other distributions. Nadarajah (2008; see also Jones, 2009, p. 72) observed that, like the Beta distribution, the Kumaraswamy distribution is a special case of McDonald’s generalized Beta of the first kind distribution (McDonald, 1984; see also McDonald and Richards, 1987). Although Naradajah only considered the case with c = 0, his observation also applies to the general case of c<bif we slightly generalize McDonald’s formulation by adding a lower- bound parameter to his density function.
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